English
If f is in L¹ and the transform is defined via an appropriate bilinear form, then the Fourier transform is a continuous function of the frequency variable.
Русский
Если f ∈ L¹ и преобразование Фурье задано через подходящую билинейную форму, то преобразование Фурье непрерывно по частоте.
LaTeX
$$$$ \\text{continuous of } w \\mapsto \\mathrm{fourierIntegral}(e, \\mu, L, f)(w). $$$$
Lean4
theorem fourierIntegral_add (he : Continuous e) (hL : Continuous fun p : V × W ↦ L p.1 p.2) {f g : V → E}
(hf : Integrable f μ) (hg : Integrable g μ) :
fourierIntegral e μ L (f + g) = fourierIntegral e μ L f + fourierIntegral e μ L g :=
by
ext1 w
dsimp only [Pi.add_apply, fourierIntegral]
simp_rw [smul_add]
rw [integral_add]
· exact (fourierIntegral_convergent_iff he hL w).2 hf
· exact (fourierIntegral_convergent_iff he hL w).2 hg
